Place the chicken, salsa verde, 1 cup green enchilada sauce, 1/4 cup water, and the can of green chiles into the Instant Pot.
Close the lid and set the valve to sealing. Cook on Manual/High pressure for 8 minutes.
When the timer finishes, allow a natural pressure release for 10 minutes, then carefully quick-release any remaining pressure and open the pot.
Remove the chicken and reserve 1/2 cup of the cooking liquid, discarding the rest.
Shred the chicken with two forks or a stand mixer paddle, then mix in the reserved 1/2 cup cooking liquid and set aside.
Spray a high-sided round or springform pan that fits inside your Instant Pot with cooking spray. Cut the 12 corn tortillas in half.
Assemble the casserole: place a layer of tortilla halves on the bottom, spoon some of the remaining 1 cup enchilada sauce over them, add a layer of shredded chicken, then sprinkle with some cheese.
Repeat the tortilla, sauce, chicken, and cheese layers one more time. Top with a final layer of tortilla halves, remaining enchilada sauce, and the rest of the cheese. Cover tightly with nonstick-sprayed foil.
Add 1/2 to 3/4 cup water to the bottom of the Instant Pot and place the trivet inside. Use a foil sling or trivet handles to carefully lower the covered pan onto the trivet.
Close the lid, set the valve to sealing, and cook on Manual/High pressure for 12 minutes (add 2–3 minutes if chilled, 4–5 minutes if frozen).
When cooking finishes, carefully quick-release the pressure. Let the casserole sit in the pot for 5–10 minutes to set before carefully removing and serving.